Proof. Bott periodicity gives \(\pi_8O\cong\pi_9O\cong\mathbb Z/2\) (Bott 1959, Corollary to Theorem II, equation (1.5)). The stable \(J\)-homomorphisms in both degrees are injective (Adams 1966, Theorem 9.5). We also use \[\pi_9^S\cong(\mathbb Z/2)^3, \qquad \bigl(\pi_{10}\operatorname{Wh}^{\mathrm{diff}}(*)\bigr)_{(2)} \cong\mathbb Z/8\oplus(\mathbb Z/2)^2.\] The first value appears in (Isaksen et al. 2023, Table 1, degree 9); the second is Rognes’s computation (Rognes 2002, Theorem 5.8), also recorded in (Rognes 2014, Theorem 3.3). Only the two-primary assertion is used.

In positive degrees \(\pi_iG=\pi_i^S\). One may use either based or unbased sphere equivalences here, since the evaluation fibration has base a sphere whose connectivity tends to infinity under stabilization. The relevant part of the long exact sequence of (5) is \[ \pi_{10}\mathbf T\longrightarrow\pi_{10}BG \xrightarrow{\partial}\pi_9H^s(*) \longrightarrow\pi_9\mathbf T\xrightarrow{\gamma_*}\pi_9BG. \tag{7}\] The image of \(\pi_{10}BO\) in \(\pi_{10}BG=\pi_9^S\) is the order-two image of \(J_9\). Theorem 2 puts it in \(\ker\partial\). Since \(\pi_9^S\) has order eight, \(\lvert\mathop{\mathrm{im}}\partial\rvert\leq4\). On the other hand, (4) identifies \[\pi_9H^s(*)=\pi_{10}\operatorname{Wh}^{\mathrm{diff}}(*),\] whose two-primary part has order \(32\). Thus \(\partial\) is not surjective. Exactness in (7) implies that the image in \(\pi_9\mathbf T\) is nonzero and is contained in \(\ker\gamma_*\), proving (6). Finally, \(\pi_9BO\to\pi_9BG\) is \(J_8\), which is injective by the other degree of Adams’s theorem cited above. ◻

Proof. A map from the compact sphere to the mapping telescope (3) has bounded height. A finite initial portion of the telescope retracts onto its last stage. Hence \(a\) is represented by a based map \(S^9\to T_{m,m}\) at a finite stage. Further stabilization preserves its stable class and makes \(m\) arbitrarily large.

Choose an oriented disk in \(B\) whose closure avoids the two critical points of \(\theta\). Collapsing the complement of its interior gives a degree-one map \(B\to S^9\), constant at the target basepoint off that disk. Equivalently one can take a smooth map of degree one supported there, obtained by a relative smooth approximation of this collapse map. Precomposition preserves the based homotopy class, since a degree-one self-map of \(S^9\) is based homotopic to the identity. The resulting family is standard on neighborhoods of both critical points.

Apply the relative angular approximation described above, retaining the standard function on smaller such neighborhoods. More explicitly, approximate the continuous map into \(C^1(S^{2m-1})\) by a finite smooth partition of unity in \(b\) multiplying smooth angular functions; then multiply its difference from \(q_{m,m}\) by a parameter cutoff which is zero on the smaller neighborhoods and one outside the original standard neighborhoods. The approximation remains uniformly close in angular \(C^1\) norm. The positive regularity margin and straight interpolation show that its stable class is still \(a\) and its spherical class is zero.

For this smooth family, the continuous positive function \(\lVert 2h_b(u)u+\nabla_Sh_b(u)\rVert\) has a positive minimum and a finite maximum on \(B\times S^{2m-1}\). Formula (2) gives (8), including the origin. Finally \(d_bg_b(ru)=r^2d_bh_b(u)\). Differentiating in \(w\) gives a degree-one expression whose angular coefficients are bounded by compactness, proving (9). The degree-two value bound and degree-one derivative bound also prove the asserted \(C^1\) extension at zero. As before, the nonzero based stable class cannot be freely nullhomotopic, because the fundamental group action fixes zero. ◻

Proof. Choose a radius \(a>0\) inside the common quadratic neighborhood and a radius \(R_0\) with an inner collar on which the family is already homogeneous. Take a small uniform \(\delta>0\) and write \[M_-^b=\{f_b\leq-\delta\},\qquad M_+^b=\{f_b\leq\delta\}.\] We first show that \(M_-^b\) is homotopy equivalent to the negative domain at infinity. On its compact part \(\lVert w\rVert\leq R_0\), choose a descending field, radial outward near \(\lVert w\rVert=R_0\). This boundary prescription is possible because homogeneity gives \[df_b(\partial_r)=2f_b/r<0\] there. Interpolate with negative gradient away from that collar. The conditions are strict linear inequalities, so these choices can be patched smoothly, also over \(b\).

The field cannot leave through \(f_b=-\delta\). It must reach the outer sphere in finite time: on the compact region its rate of descent has a positive lower bound, while \(f_b\) has a finite lower bound. The exit is transverse. Stopping the flow at its first exit and fixing all points outside the ball gives a deformation retraction onto \[M_-^b\cap\{r\geq R_0\}.\] The globally stopped homotopy is continuous, which is sufficient here. If \(h_b\) denotes the angular function at infinity, angular projection of this outer portion has, over \(u\) with \(h_b(u)<0\), the radial fiber \[\left[\max\left\{R_0,\sqrt{\frac{\delta}{-h_b(u)}}\right\},\infty\right).\] Contracting each such interval to its continuous lower endpoint proves the homotopy equivalence. The open and closed negative domains have the same homotopy type, since zero is a regular value. In particular, \[ M_-^b\simeq S^{k-1}. \tag{11}\]

On \(f_b\geq\delta\), normalized negative gradient has \(df_b\)-derivative \(-1\) and bounded speed. Indeed the gradient is bounded away from zero on the compact part and grows at least linearly at infinity. Extend this to a bounded smooth field by cutting off below \(\delta\), away from zero. Each point with \(f_b>\delta\) reaches the level \(\delta\) after time \(f_b-\delta\). Stopped flow gives a deformation retraction of \(\mathbb R^{k+l}\) onto \(M_+^b\), so \[ M_+^b\simeq *. \tag{12}\]

We give the local handle argument to justify its use for these noncompact sublevels. If the Morse index is \(j\), choose local linear coordinates with \(Q_b=-\lVert x\rVert^2+\lVert y\rVert^2\), where \(x\in\mathbb R^j\). The coordinate changes and their inverses have uniformly bounded norms; a global frame is not needed. Choose a smooth function \(\mu\) on \([0,\infty)\) with \[-1<\mu'\leq0,\qquad \mu(0)>\delta, \qquad \mu(s)=0\text{ for }s\geq2\delta.\] Modify the function in this neighborhood to \[\widetilde f_b=f_b-\mu(\lVert x\rVert^2+2\lVert y\rVert^2).\] The modification is supported where \(f_b<\delta\). Its derivatives in the chart are \[-2(1+\mu')x,\qquad 2(1-2\mu')y.\] Thus zero is its only critical point and has value below \(-\delta\). We have \(\{\widetilde f_b\leq\delta\}=M_+^b\). Bounded normalized descent on the slab between \(-\delta\) and \(\delta\) retracts this set, relative to \(M_-^b\), onto \(\{\widetilde f_b\leq-\delta\}\).

The latter set retracts, relative to \(M_-^b\), onto \[M_-^b\cup D,\qquad D=\{y=0,\ \lVert x\rVert^2\leq\delta\}.\] For membership of \(D\), note that \(s+\mu(s)\geq\mu(0)>\delta\) for \(0\leq s\leq\delta\). At an added point outside \(M_-^b\), shrink \(y\) radially to zero if \(\lVert x\rVert^2\leq\delta\), and otherwise to length \(\sqrt{\lVert x\rVert^2-\delta}\). The modified function decreases during this motion, and it agrees continuously with fixing \(M_-^b\) at the boundary. This proves the claimed relative deformation.

The relative homology of \((M_+^b,M_-^b)\) is therefore that of one \(j\)-disk modulo its boundary. Equations (11)–(12) force \(j=k\), and the boundary map makes \(\partial D\) a generator of \(H_{k-1}(M_-^b;\mathbb Z)\). Within the local negative cone, this sphere expands to \(aS(E_b^-)\), where \(E_b^-\) is the negative eigenspace in the original Euclidean metric.

Flow this sphere to the outer sphere using the descending field above and project angularly. The construction varies smoothly with \(b\). On each fiber it takes a homology generator to a generator in the negative domain. Both source and target have the homotopy type of the simply connected sphere \(S^{k-1}\). They have CW type, so the integral homology isomorphism is a homotopy equivalence by the homological Whitehead theorem (Hatcher 2002, Corollary 4.33). The fiberwise criterion for maps of bundles over a CW base makes it a fiber homotopy equivalence. This identifies the claimed spherical fibration. ◻

Proof. There is \(c_0>0\) such that \(\lVert\nabla_WG_0\rVert\geq c_0\lVert W\rVert\). For \(W\ne0\) choose a smooth cutoff \(\rho_\eta\) which vanishes on \([0,\eta]\), equals one on \([2\eta,\infty)\), and has derivative bounded by \(C/\eta\), and set \[G_\eta(b,W)=g_b(w)\rho_\eta\left(\frac{\lVert w\rVert}{\lVert W\rVert}\right)+q'(z).\] The potentially nonsmooth term vanishes in an open cone around \(w=0\). Elsewhere it is smooth, so \(G_\eta\) is smooth away from \(W=0\). On the support of \(1-\rho_\eta\), we have \(\lVert w\rVert\leq2\eta\lVert W\rVert\). Homogeneity and the angular derivative bound give \[ \lvert G_\eta-G_0\rvert\leq C_2\eta^2\lVert W\rVert^2, \qquad \lVert d_WG_\eta-d_WG_0\rVert\leq C_2\eta\lVert W\rVert. \tag{13}\] In the second estimate the derivative of \(\lVert w\rVert/\lVert W\rVert\) costs at most \(\lVert W\rVert^{-1}\) where the cutoff changes. For small \(\eta\), the straight homotopy from \(G_0\) to \(G_\eta\) retains a linear nonzero-gradient bound.

Let \(\alpha_A\) vanish for \(\lVert W\rVert\leq A\), equal one for \(\lVert W\rVert\geq2A\), and have derivative bounded by \(C_3/A\). Replace \(F\) by \[F_A=(1-\alpha_A)F+\alpha_AG_\eta.\] On the transition annulus its derivative differs from \(d_WG_0\) by at most \[M_1+C_2\eta\lVert W\rVert +\frac{C_3}{A}\bigl(M_0+C_2\eta^2\lVert W\rVert^2\bigr).\] Relative to \(\lVert W\rVert\) this is \(O(\eta+\eta^2+A^{-1})\). Choose \(\eta\) small and then \(A\) large so that the bound is less than \(c_0\lVert W\rVert/2\). No critical points occur in the annulus or its exterior. The original critical points already lie in a uniform ball, since \(d_WF\) differs from a linearly bounded-below gradient by at most \(M_1\). Taking \(A\) beyond that ball preserves their neighborhoods. The function is unchanged near zero and thus smooth everywhere. ◻

Proof. By (15), near the zero set a field can make \(df_b\) and \(ds_b\) both strictly positive, or both strictly negative. The same holds at the two side levels for small uniform \(\epsilon\). On an end collar write \(s_b=h\), angular and independent of \(t\), and use \[V=\partial_t+h\nabla_Yh.\] Then \[dt(V)=1,\quad ds_b(V)=h\lVert\nabla_Yh\rVert^2,\quad df_b(V)=h\bigl(2rr'+r^2\lVert\nabla_Yh\rVert^2\bigr).\] These give the required signs, including strictness on the side levels, since \(dh\ne0\) near \(h=0\). Away from the prescribed regions, use strict signed gradient ascent or descent for \(f_b\). Convex patching preserves every applicable strict linear inequality, and can keep the displayed prescription on smaller end collars. The construction is smooth over the compact parameter space. Fields on cornered regions are understood to extend locally across their faces. ◻

Proof. For \(V_{-,b}\) the inner face and side face are entry faces, and the outer face is the only exit. Every trajectory exits in finite forward time: on the compact region \(P_{-,b}\), \(-df_b(V_{-,b})\) has a positive lower bound and \(f_b\) has bounded range. Backwards, the same argument forces a trajectory to reach an entry face.

Flow to the first intersection with \(t=1\) maps each entry face smoothly and injectively into \(T_{-,b}^1\). Transversality to the exit face gives smooth dependence of the exit time, also at the side’s outer edge where the time is zero. Extensions across the corner faces show that these maps have full rank up to the edges. Their images meet precisely along the shared inner edge and otherwise are disjoint, by uniqueness of flow; backwards flow gives their union as the entire outer tube. The image of the inner face lies in the interior of the outer tube, since the field points strictly inward from the side. The band identification (16) identifies that side with \(C_b\) and proves the stated decomposition.

Choose \(\delta>0\) as in Lemma 6, also small enough that the sphere \(a(S^{k-1}\times\{0\})\) in the fixed negative eigenspace lies in \(f_b\leq-\delta\). This sphere is the core of the inner tube. Its image under the negative flow remains in this sublevel. The core represents a generator there by that lemma, and therefore its image at \(R_0\) represents a generator in the outer radial portion. Angular projection carries it to a generator in the open negative domain. Inclusion of \(T_{-,b}^1\) into this open domain is a homotopy equivalence for our small \(\epsilon\). Thus \(j_b\) has degree \(\pm1\) on the sphere homotopy types of its source and target. These types are simply connected, so \(j_b\) is a homotopy equivalence.

Excision with collars now gives \[H_*(C_b,\partial_-C_b;\mathbb Z) \cong H_*(T_{-,b}^1,j_b(T_-^0);\mathbb Z)=0.\] This includes connectedness of \(C_b\). The two tubes and their common boundary are simply connected, so van Kampen applied to the decomposition implies \(\pi_1C_b=0\). The cobordism is orientable, and Poincaré–Lefschetz duality gives vanishing of the relative homology at its outgoing end as well. Both boundary components are diffeomorphic to \(S^{k-1}\times S^{l-1}\) and hence simply connected. All these smooth spaces have CW type, so the homological Whitehead theorem (Hatcher 2002, Corollary 4.33) makes both boundary inclusions homotopy equivalences. Regularity and properness give a smooth bundle of these cobordisms over \(B\), including their boundary collars, with fixed incoming identification. ◻

Proof. Adjoin a fixed incoming collar to \(C_b\), obtaining \[C_b^-=(\Sigma\times[-1,0])\cup C_b.\] Its interval stabilization is \(C_b^-\times J\), with incoming \(\Sigma\times J\). In the convention of (20), the two faces \(C_b\times\partial J\) and the outgoing end times \(J\) form the movable boundary. The added incoming-collar portions supply the fixed vertical boundary when a boundary-relative model is used. Extending across \(\Sigma\times J\hookrightarrow Y\) replaces that incoming collar by \(Y\times[-1,0]\). The result is exactly (22), using (16) for the attached piece.

This construction works on transition maps as well. Local product trivializations of \(C_b\) fix its incoming collar and can be made product on outgoing collars, without fixing their outgoing end. Extend them by identity over the added collar and take their product with \(\operatorname{id}_J\). A fixed bending of the two interval coordinates gives the usual boundary-relative stabilized cylinder. Near the lower gluing seam the transitions are identity; near the upper seam they preserve both interval coordinates and act only in the remaining boundary variable. Thus fixed planar corner adjustments descend through the transitions. Extending by identity over the complement in \(Y\) gives the same family as (22). This is a single interval stabilization with both side faces movable. ◻

Proof. Before rounding, the outgoing boundary of \(W_b\) comprises the two sign domains at \(t=0\), the two band sides, and the band end at \(t=1\). Its complement is the union of \(P_{-,b}\) and \(P_{+,b}\) with the upper collar \(Y\times[1,2]\). Use Lemma 8 on the sign regions. On and above the upper end, its common angular prescription extends across the band and has \(dt>0\). It can be continued to the top with that property. The resulting field points from every outgoing face of \(W_b\) into the complement, and its only exit face is \(t=2\).

We describe a rounding transverse to this field. At a lower corner put \[v=t,\qquad w=\pm s_b-\epsilon;\] the complement is the quadrant \(v,w\geq0\). At an upper corner put \(v=t-1\) and use the same \(w\); the complement is the union \(v\geq0\) or \(w\geq0\). At either corner the field has \(dv(V)>0\) and \(dw(V)>0\). Round the interface by a curve in the first quadrant at the bottom, and in the third quadrant at the top, smoothly joining the unchanged straight parts. Choose its conormal into the complement as a nonzero nonnegative combination of \(dv,dw\). This makes the rounded interface transverse with the same orientation.

The product end coordinates allow fixed small planar roundings over all parameters. If the stabilized piece and its exterior are kept as separate pieces at the bottom seam, move their dividing wall in the small corner neighborhood so it reaches the rounded interface transversely. All transition maps there are identity. Each piece is then a fixed planar adjustment of its original collared model, so Lemma 10 still describes the rounded family.

Every forward trajectory reaches \(t=2\) in finite time. In sufficiently small lower and upper collars, \(dt(V)\) has a uniform positive lower bound; this bounds transit time there and prevents reentry across their inner thresholds in the wrong direction. In the remaining compact middle, a trajectory stays in one sign region and its signed \(f_b\) increases at a rate bounded below, with bounded range. Its time there is therefore finite. It cannot exit across the entry boundary. The same argument backwards reaches the rounded outgoing boundary of \(W_b\).

Transversality gives smooth hitting times in the initial point and in \(b\). Rescaling these times gives a product diffeomorphism of the complementary cobordism relative to its entry, say \(E_b=\partial_+W_b\). This proves a product identification over the entire possibly varying bundle \(E\to B\); it does not merely assert that the individual fibers are products.

Finally, adding \(E_b\times I\) to the outgoing end changes no incoming-relative h-cobordism bundle class. Extend the transverse field a small uniform distance into \(W\) to obtain a collar \(E\times[-\eta,0]\). A fixed increasing smooth diffeomorphism \([-\eta,1]\to[-\eta,0]\), equal to the identity near \(-\eta\), compresses the added cylinder into this collar, and is identity on the rest of \(W\). It is defined without choosing a trivialization of \(E\to B\). The enlarged total is the literal product \(B\times Y\times[-1,2]\), relative to its incoming boundary. Hence \([W]=0\). ◻

Proof. Here \(\dim\Sigma=k+l-2\geq38\), whereas the bound in (23) for \(r=10\) is \(35\). Hence \(H(\Sigma)\to H^s(\Sigma)\) is \(10\)-connected. The composite \(H(\Sigma)\to H(\Sigma\times J)\to H^s(\Sigma\times J)\) has the same connectivity: its target is the tail of the defining stabilization telescope. It therefore induces an isomorphism on \(\pi_9\).

Set \(d=\min(k,l)-1\). The source of \(e\) is \((d-1)\)-connected, and \(Y\) has vanishing homotopy groups in degrees at most \(d\). Thus \(e\) is \(d\)-connected: it is an isomorphism below degree \(d\) and a surjection in degree \(d\). The stable extension \(H^s(\Sigma\times J)\to H^s(Y)\) is consequently \((d-1)\)-connected, with \(d-1\geq18\), and also induces an isomorphism on \(\pi_9\). Compatibility of extension and stabilization identifies the composite of these two isomorphisms with (24).

The space \(H(\Sigma)\) is connected, as shown above. For a connected target, free homotopy classes of maps from \(S^9\) are orbits of based classes under the fundamental group action. The orbit of zero is just zero. Thus the same detection of nullity applies to sphere families without a specified basepoint. ◻

Completion of the proof of Proposition 5. First replace the outer negative domains by their truncated domains \(T_{-,b}^1\); moving a regular boundary through a sufficiently small angular collar is an isotopy. By Lemma 9 and (25), each of these domains is \(j_b(T_-^0)\) with a smoothly varying product collar attached. Shrink that collar: if its product coordinate is \([0,1]\), retain only \([0,s]\) and let \(s\) decrease to zero. The outer boundaries vary through embeddings, including at \(s=0\), where they equal \(\partial j_b(T_-^0)\). Thus the family is homotopic as tube domains to \(j_b(T_-^0)\).

Fix \(T_-^0\cong S^{k-1}\times D^l\) and let \[e_b:S^{k-1}\hookrightarrow Y,\qquad e_b(u)=j_b(u,0)\] be its cores. We may replace \(j_b(T_-^0)\) by small round normal neighborhoods of \(e_b\). Here is a precise version that removes framing choices. First shrink its disk factors to a uniformly small radius. In metric normal-bundle charts of the cores write \[j_b(u,z)=\bigl(\psi_b(u,z),\xi_b(u,z)\bigr),\qquad \psi_b(u,0)=u,\quad \xi_b(u,0)=0.\] On this smaller domain use, for \(0<t\leq1\), the embeddings \[ (u,z)\longmapsto \bigl(\psi_b(u,tz),\ t^{-1}\xi_b(u,tz)\bigr). \tag{26}\] These are compositions of the original embedding with dilation in the disk and normal directions. The estimates \(\lVert\xi_b(u,tz)\rVert\leq Ct\lVert z\rVert\) keep them in the chosen chart. At \(t=0\) the formula extends smoothly to \((u,D_z\xi_b(u,0)z)\). The normal derivative is an isomorphism, so the limit is also an embedding. Its fiber ellipsoids can be changed to round balls by interpolation of positive definite metrics, with the common disk radius chosen sufficiently small. At the level of domains this requires no fixed normal framing.

The core family is homotopic through embeddings to the fixed standard core. Indeed, as maps into \(Y=S^{k+l-1}\), both it and the constant standard family are nullhomotopic because \[\dim(B\times S^{k-1})=k+8<k+l-1.\] Choose a smooth homotopy, constant on collars of its two time endpoints. Its parameter dimension, including time, is ten. Relative fiberwise multijet transversality avoids double points since \[10+2(k-1)<k+l-1.\] Vertical rank-defect strata have codimension at least \(l+1>10+(k-1)\) and are avoided as well. Here we use the parametric multijet theorem in the form of (Borodzik et al. 2025, Theorem 2.3.4 and Corollary 2.3.5(iii)), and explain its compact-parameter relative application. Cover the compact parameter manifold \(B\times[0,1]\), away from smaller endpoint collars, by finitely many coordinate charts. In each chart multiply the local perturbations by a parameter cutoff supported away from the endpoint collars. Local source cutoffs allow independent variations at distinct source points, as required for the two-point jets. Apply transversality successively on compact subsets covering these charts; sufficiently small subsequent perturbations preserve avoidance already obtained on compact subsets. The rank estimate first gives fiberwise immersions. Compactness and \(C^1\) openness then give a uniform neighborhood of the diagonal on which distinct sufficiently close source points remain distinct. Apply the two-point argument on the compact complement of this neighborhood. The strict dimension bounds make each transverse bad-stratum preimage empty. The endpoint families remain fixed, and their embedding property persists on smaller endpoint collars by \(C^1\) openness. This proves that every member of the resulting homotopy is an embedding. Uniformly small round normal neighborhoods follow this family to a fixed tube.

All the resulting homotopies of domains give homotopies of tube data. Use boundary collars for local defining-function choices; the fibers of these choices are convex, so partitions of unity in parameters give global choices. Continuous families in the smooth topology suffice for the homotopy class; when smooth parameter dependence is needed, relative approximation preserves embeddings and regularity on compact parameter sets. We have proved nullity of the enlarged family in \(T_{k,l}\). Adding the missing negative squares returns to the diagonal system and proves the proposition. ◻

Proof. Let \(\varphi_t\) be the assumed original flow, generated by \(H_t\). The path \(\varphi_{1-t}\circ\varphi_1^{-1}\) starts at the identity, ends at \(\varphi_1^{-1}\), and is generated by \[ H_t^Q=-H_{1-t}. \tag{37}\] Indeed differentiation of this path gives \(-X_{H_{1-t}}\) at its current point; the fixed composition on the right introduces no conjugation of the Hamiltonian.

Choose \(0\le\zeta\le1\) supported in \((-\nu,\nu)^2\) and identically \(1\) on \((-\nu_c,\nu_c)^2\). Choose \(0\le\xi\le1\) of compact support in \(\mathbb R^2\), with \(\xi=1\) on a ball to be specified. On \(T^*U\) set \[ \widetilde H_t(q,\alpha;p_q,p_\alpha) =H_t^Q(q,p_q)\zeta(\alpha)\xi(p_\alpha), \tag{38}\] and extend by zero to \(T^*\mathbb R^d\). This is smooth with uniform compact support: \(Q\) is compact, \(H_t^Q\) has uniformly compact momentum support, and both additional coordinate pairs are cut off.

The convention \(\iota_{X_H}d\lambda=-dH\) gives the radial equations \[ \dot\alpha=H_t^Q\zeta\,\nabla\xi(p_\alpha), \qquad \dot p_\alpha=-H_t^Q\nabla\zeta\,\xi(p_\alpha). \tag{39}\] Set \[M=\sup_{t,q,p_q}\lvert H_t^Q(q,p_q)\rvert\,\sup_\alpha\lVert\nabla\zeta(\alpha)\rVert.\] This number is independent of the size of the plateau of \(\xi\). Take \(\xi=1\) on \(\{\lVert p_\alpha\rVert\le M+1\}\). A trajectory starting on \(\mathcal L_0\) starts with \(p_\alpha=0\), and the second equation gives \(\lVert p_\alpha(t)\rVert\le Mt\) for \(0\le t\le1\). Hence it stays in this plateau, the first equation vanishes, and \(\alpha\) is constant. When \(\alpha\in(-\nu_c,\nu_c)^2\), also \(\nabla\zeta=0\); then \(p_\alpha=0\) and the \(T^*Q\) motion is exactly the reverse isotopy. It sends \(L\) to \(Q_0\). Since all radial positions on the entire cylinder are fixed, no other part of that cylinder enters the central slab. This proves (36). ◻

Proof. Require \(\lVert D_xX-I\rVert<\varepsilon<1\) uniformly. For each fixed \(p\), the map \(x\mapsto X(x,p)\) is a local diffeomorphism and satisfies \[\lVert X(x_1,p)-X(x_2,p)\rVert\ge(1-\varepsilon)\lVert x_1-x_2\rVert.\] It is injective and proper, since it is the identity outside a compact set. Its image is both open and closed in \(\mathbb R^d\), so it is onto. The inverse \(x=x(X,p)\) is smooth jointly in \((X,p)\).

For a step from time \(a\) to time \(b\), let \(\varphi_{s,a}\) denote its partial flow. Cartan’s formula and the sign convention give \[\mathcal L_{X_{\widetilde H_s}}\lambda =d\bigl(\lambda(X_{\widetilde H_s})-\widetilde H_s\bigr).\] Consequently the action integral \[ A(x,p)=\int_a^b \bigl(\lambda(X_{\widetilde H_s})-\widetilde H_s\bigr) (\varphi_{s,a}(x,p))\,ds \tag{41}\] satisfies \(\psi^*\lambda-\lambda=dA\). It is compactly supported: outside \(K\) the trajectory is stationary and its integrand is zero. On the graph, expressed in mixed coordinates, put \[S(X,p)=A(x(X,p),p)+x(X,p)\cdot p.\] Then \[dS=P\cdot dX+x\cdot dp.\] Writing \(S=X\cdot p+\sigma\) gives (40). Explicitly, \[ \sigma(X,p)=A(x(X,p),p)+(x(X,p)-X)\cdot p. \tag{42}\] This expression vanishes outside the compact mixed-coordinate image of \(K\). Its spatial projection lies in a fixed compact subset of \(U\), because both the input and output trajectories remain in \(K\). Enlarge that projection slightly to obtain \(D\Subset U\). Thus for every \(X\in U\setminus D\), \(\sigma\) and all its derivatives vanish on a neighborhood of \(X\), for all momenta. Finally \(I+\sigma_{pX}=D_Xx\) is invertible, and \(\sigma_p=x-X\) tends uniformly to zero with the length of the time increment. ◻

Proof. Introduce \(z,p\in\mathbb R^d\). Before choosing cutoffs, consider the local expression \[F_o(X+z,w)+\sigma(X,p)-z\cdot p, \qquad X+z\in U.\] Its \(p\)- and \(z\)-equations are \(z=\sigma_p(X,p)\) and \(p=\nabla_xF_o(X+z,w)\). Together with the old \(w\)-equation, these are precisely the mixed graph equations for composition with \(\psi\): \(z\) is the displacement from the new base point \(X\) to the old one, and \(p\) is the old momentum. The cutoffs below keep the shifted base point in \(U\) and switch off the shift for large \(\lVert w\rVert\). We will then compare the result with \(g_o(X,w)-z\cdot p\) to restore homogeneity at infinity.

Choice of cutoffs. Choose \(\zeta_1:U\to[0,1]\) of compact support, identically \(1\) on an open set \(V\) containing \(D\). Fix \(\delta_0>0\) so that \[ 2\delta_0<\operatorname{dist}(\operatorname{supp}\zeta_1,\partial U), \qquad \lVert\sigma_p\rVert_\infty<\delta_0. \tag{44}\] The first choice is made before refining the subdivision to obtain the second. Let \(\beta_1:\mathbb R^d\to[0,1]\) be \(1\) for \(\lVert z\rVert\le\delta_0\) and \(0\) for \(\lVert z\rVert\ge2\delta_0\). Choose a smooth radial \(\chi:\mathbb R^r\to[0,1]\), equal to \(1\) on a ball strictly larger than \(\{\lVert w\rVert\le R_o\}\), of compact support, and such that \[ \lVert\nabla\chi(w)\rVert\le\delta_1/\lVert w\rVert\quad(w\ne0). \tag{45}\] Here \(\delta_1>0\) may be arbitrarily small. For example, after fixing the inner radius \(R_1>R_o\), take \(\chi(w)=\vartheta(\log(\lVert w\rVert/R_1)/T)\) outside that radius, where \(\vartheta\) is a fixed smooth cutoff equal to \(1\) near \((-\infty,0]\) and \(0\) near \([1,\infty)\); choosing \(T\) sufficiently large gives (45). Extend it by \(1\) at smaller radii.

Define \[\begin{align*} \kappa(X,z,w)&=\zeta_1(X)\beta_1(z)\chi(w), &x&=X+\kappa(X,z,w)z,\tag{46}\\ \widehat F(X,w,z,p) &=F_o(X+\kappa(X,z,w)z,w)+\sigma(X,p)-z\cdot p. \tag{47}\end{align*}\] The shifted argument always lies in \(U\). Indeed, when the shift is nonzero, \(X\in\operatorname{supp}\zeta_1\) and its length is less than \(2\delta_0\); the buffer in (44) places that entire ball in \(U\). When the shift vanishes the assertion is immediate.

The full critical equations. Write \(a=\nabla_xF_o(x,w)\), identifying Euclidean covectors and vectors. Direct differentiation, before discarding any cutoff terms, gives \[\begin{align*} \widehat F_p&=\sigma_p(X,p)-z,\tag{48}\\ \widehat F_z&=\kappa a+(a\cdot z)\nabla_z\kappa-p, \tag{49}\\ \widehat F_w&=\nabla_wF_o(x,w)+(a\cdot z)\nabla_w\kappa, \tag{50}\\ \widehat F_X&=a+(a\cdot z)\nabla_X\kappa+\sigma_X(X,p). \tag{51}\end{align*}\] At any vertical critical point, (48) forces \(z=\sigma_p(X,p)\), hence \(\lVert z\rVert<\delta_0\) and \(\beta_1=1\) on a neighborhood of that point. When \(\lVert w\rVert\ge R_o\), the old function is homogeneous at every allowed shifted base point. Equations (43) and (45) therefore imply \[ \lVert\widehat F_w\rVert \ge(c_o-C_o\delta_0\delta_1)\lVert w\rVert. \tag{52}\] Choose \(\delta_1\) so that \(C_o\delta_0\delta_1<c_o/2\). Thus no vertical critical point has \(\lVert w\rVert\ge R_o\); at every one of them \(\chi=1\) on a neighborhood as well.

There are two open regions to consider, namely \(V\) and \(U\setminus D\); they cover \(U\). For \(X\in V\), the three cutoff factors are \(1\) near a solution, and the critical equations become \[ \nabla_wF_o(x,w)=0, \qquad p=\nabla_xF_o(x,w), \qquad x=X+\sigma_p(X,p), \qquad x=X+z. \tag{53}\] They express exactly composition with the graph (40), and the generated covector is \[ \widehat F_X=p+\sigma_X(X,p)=P. \tag{54}\] The critical equation is regular here. In local total coordinates \((x,w,p,X)\), first impose the regular old equation \(\nabla_wF_o=0\). With \((x,w)\) fixed, the equation \(p-\nabla_xF_o=0\) has identity coefficient in \(p\). With \((x,w,p)\) fixed, the last graph equation has invertible coefficient \(I+\sigma_{pX}\) in \(X\). This triangular order proves surjectivity of the differential of the full new vertical-gradient map.

For \(X\in U\setminus D\), the function \(\sigma\) vanishes on a spatial neighborhood for all \(p\). The equations instead reduce to \[ z=0,\qquad x=X,\qquad \nabla_wF_o(X,w)=0,\qquad p=\zeta_1(X)\nabla_xF_o(X,w). \tag{55}\] The auxiliary variable \(p\) may therefore differ from the old momentum. Nevertheless, (51) gives exactly \[ \widehat F_X=\nabla_xF_o(X,w), \tag{56}\] because all the spatial cutoff terms are multiplied by \(z=0\). This is the identity correspondence, which is the true graph of \(\psi\) over this region. For regularity, first use the equation \(-z=0\) to eliminate \(z\), then the equation \(\zeta_1\nabla_xF_o-p=0\) to eliminate \(p\); the remaining equation is the regular old vertical-gradient equation. This also checks the entire transition region of \(\zeta_1\). The two descriptions agree on the overlap, where \(\zeta_1=1\).

The correspondence of critical loci. For an old critical point \((x,w)\) put \(p_o=\nabla_xF_o(x,w)\) and \((X,P)=\psi(x,p_o)\). Define \[ (x,w)\longmapsto \bigl(X,w,z=x-X,\ p=\zeta_1(X)p_o\bigr). \tag{57}\] If \(X\in V\), these are the mixed graph equations, with \(p=p_o\). If \(X\notin D\), the true graph is the identity, so \(x=X\), \(z=0\), and (55) applies. Thus (57) is a smooth map into \(\operatorname{Crit}(\widehat F)\) on all of \(\operatorname{Crit}(F_o)\). Its inverse is \[ (X,w,z,p)\longmapsto(X+\kappa(X,z,w)z,w). \tag{58}\] The preceding equations show that these maps are inverse and that the generated image is exactly \(\psi(\mathcal L_o)\). The step preserves \(T^*U\), since its support is compactly contained there. Thus no old or new critical point is lost at the base boundary, and \(\Gamma_{\widehat F}\) is a diffeomorphism onto the transported image. All new solutions obey a uniform bound: \(\lVert w\rVert<R_o\), \(\lVert z\rVert<\delta_0\), and \[\lVert p\rVert\le \sup\{ \lVert\nabla_xF_o(x,w)\rVert:x\in\overline U, \lVert w\rVert\le R_o\}<\infty.\]

Restoring exact homogeneity at infinity. Set \(W=(w,z,p)\) and consider the homogeneous reference \[ G_0(X,W)=g_o(X,w)-z\cdot p. \tag{59}\] It is \(C^1\) everywhere, homogeneous of degree two, and satisfies \[ \lVert\nabla_WG_0\rVert^2 =\lVert\nabla_wg_o\rVert^2+\lVert z\rVert^2+\lVert p\rVert^2 \ge\min(c_o^2,1)\lVert W\rVert^2. \tag{60}\] Both the value error and its full vertical first derivative are uniformly bounded. To verify this explicitly, write \[\begin{align*} E(X,W)&=\widehat F(X,W)-G_0(X,W)\\ &=\bigl[F_o(X+\kappa z,w)-F_o(X,w)\bigr] +\bigl[F_o(X,w)-g_o(X,w)\bigr]+\sigma(X,p). \tag{61}\end{align*}\] The first bracket and every term in its vertical derivative can be nonzero only with \(w,z\) in fixed bounded sets, where the required derivatives of \(F_o\) are uniformly bounded. The second bracket and its vertical derivative are bounded because it vanishes for \(\lVert w\rVert\ge R_o\). The final term has compact support. The expression \(-z\cdot p\), although unbounded, has cancelled exactly. Thus for some finite \(M_0,M_1\), \[ \lvert E(X,W)\rvert\le M_0, \qquad \lVert\nabla_WE(X,W)\rVert\le M_1 \quad(X\in\overline U, W\in\mathbb R^{r+2d}). \tag{62}\] No compact-support assertion for \(E\) in all its variables is needed.

Apply Lemma 7 with \(q'(z,p)=-z\cdot p\), uniformly over \(\overline U\). Its homogeneous smoothing is \[ G_\eta(X,W)= \rho_\eta(\lVert w\rVert/\lVert W\rVert)g_o(X,w)-z\cdot p \quad(W\ne0), \tag{63}\] where \(\rho_\eta\) vanishes for arguments at most \(\eta\) and is \(1\) for arguments at least \(2\eta\). Choose \(\eta\) small and then \(A\) sufficiently large as in the lemma, with \(A\) beyond the uniform critical-locus bound. Let \(\tau_A\) be the radial cutoff from that construction and set \[ F_n=(1-\tau_A)\widehat F+\tau_A G_\eta. \tag{64}\] The lemma shows that this function is smooth, equals \(G_\eta\) for \(\lVert W\rVert\ge2A\), and has no new critical points. It leaves a neighborhood of the entire critical locus unchanged, so \(F_n\) has the same regular critical locus and generated parametrization as \(\widehat F\). It also supplies a straight homotopy from \(G_0\) to \(G_\eta\) through regular \(C^1\) homogeneous functions.

The new homogeneous function \(G_\eta\) is smooth off zero, smoothly dependent on \(X\) up to \(\overline U\), and has a uniform positive gradient margin on the unit sphere. Homogeneity and compactness there give the next constants of the form (43); the remaining uniform smooth bounds follow directly from the construction. The quadratic form \(-z\cdot p\) is nondegenerate with signature \((d,d)\): in fixed linear coordinates \(z=a+b\), \(p=a-b\) it equals \(-\lVert a\rVert^2+\lVert b\rVert^2\). The space of nondegenerate forms of signature \((d,d)\) is connected. Choose a fixed path in this space from \(-z\cdot p\) to the standard split form in the chosen added coordinates. Its direct sum with \(g_o(X,w)\) has nonzero total gradient off the origin, since both block gradients can vanish only there. This path is independent of \(X\), so direct sum defines a homotopy over the whole base. Together with the straight homotopy to \(G_\eta\), it preserves the free stable tube class on every fixed compact parameter slice. This proves all the assertions. ◻

Proof of Theorem 1. Take the class \(a\ne0\) with \(\gamma_*a=0\) supplied by Proposition 3, and the closed exact embedded Lagrangian \(L\subset T^*(S^9\times S^{N-1})\) supplied by Proposition 14. Suppose that a Hamiltonian isotopy as in the theorem sent the zero section to \(L\). Lemma 15 supplies a compactly supported Euclidean Hamiltonian whose final cylinder is the zero graph over \(U_c\). Start with (34) and apply Proposition 17 to each of its \(s\) increments. Only finitely many steps are required. The subdivision is fixed by the mixed-coordinate and spatial-buffer conditions, which depend on the Hamiltonian and its support. At each step the new family constants change only the logarithmic cutoff, smoothing parameter and outer radius; they impose no further restriction on the step length. Thus the estimates and cutoff radii may be chosen anew at each step. We obtain a smooth family \[F^{(s)}:U\times\mathbb R^{2p}\longrightarrow\mathbb R, \qquad p=m+ds,\] with exactly homogeneous smooth-off-zero data at infinity, whose regular critical locus maps diffeomorphically onto \(\widetilde\varphi_1(\mathcal L_0)\).

Over \(U_c\), the full critical-locus projection \[ \operatorname{Crit}(F^{(s)})\big|_{U_c}\longrightarrow U_c \tag{65}\] is a diffeomorphism: it is the generating parametrization followed by projection of the zero graph. In particular there is precisely one critical point over every \(X\in U_c\), and these points form a smooth section. Their vertical Hessians are invertible. Indeed, if \(v\in\mathbb R^{2p}\) were in the kernel of such a Hessian, then \[D(\partial_WF^{(s)})_{(X,W)}(0,v) =\partial_W^2F^{(s)}(X,W)v=0.\] Since the full critical equation is regular, its zero set has this kernel as its tangent space. Thus \((0,v)\) would be tangent to \(\operatorname{Crit}(F^{(s)})\) and in the kernel of the differential of (65). That projection is a diffeomorphism, so \(v=0\); the square Hessian is nonsingular.

Fix \(v_0\in S^{N-1}\) and restrict the base to \[b\longmapsto\iota((b,v_0),0),\qquad b\in S^9.\] The resulting smooth family \(f_b:\mathbb R^{2p}\to\mathbb R\) therefore has exactly one nondegenerate fiber critical point, varying smoothly with \(b\). Its homogeneous data belong to \(T_{p,p}\), since each transport step added signature \((d,d)\). On this fixed slice the initial homogeneous family was \(g_b\), and every subsequent step preserved its class after split stabilization. Hence, writing \(g_\infty:S^9\to\mathbf T\) for the final stable tube family, \[[g_\infty]_{\mathrm{free}}=[a]_{\mathrm{free}}\ne[*], \qquad \gamma\circ g_\infty\simeq *.\] This formulation does not require the smoothing homotopies to fix a chosen basepoint. Equivalently, the basepoint paths supplied by these homotopies identify the resulting based class with \(a\). All hypotheses of Proposition 5 hold, including the uniform homogeneous behavior and the required large indices. That proposition makes the same tube family freely null in the stable tube space, contradicting the non-null class of Proposition 3. The assumed compactly supported Hamiltonian isotopy does not exist. This closed exact embedded \(L\) is therefore the required counterexample. ◻